Kampé de Fériet function

In mathematics, the Kampé de Fériet function is a two-variable generalization of the generalized hypergeometric series, introduced by Joseph Kampé de Fériet.

The Kampé de Fériet function is given by

${}^{p+q}f_{r+s}\left( \begin{matrix} a_1,\cdots,a_p\colon b_1,b_1{}';\cdots;b_q,b_q{}'; \\ c_1,\cdots,c_r\colon d_1,d_1{}';\cdots;d_s,d_s{}'; \end{matrix} x,y\right)= \sum_{m=0}^\infty\sum_{n=0}^\infty\frac{(a_1)_{m+n}\cdots(a_p)_{m+n}}{(c_1)_{m+n}\cdots(c_r)_{m+n}}\frac{(b_1)_m(b_1{}')_n\cdots(b_q)_m(b_q{}')_n}{(d_1)_m(d_1{}')_n\cdots(d_s)_m(d_s{}')_n}\cdot\frac{x^my^n}{m!n!}.$

Applications

The general sextic equation can be solved in terms of Kampé de Fériet functions.^[1]

References

1. Mathworld - Sextic Equation

• Exton, Harold (1978), Handbook of hypergeometric integrals, Mathematics and its Applications, Chichester: Ellis Horwood Ltd., ISBN 978-0-85312-122-0, MR 0474684
• Kampé de Fériet, M. J. (1937), La fonction hypergéométrique., Mémorial des sciences mathématiques (in French) 85, Paris: Gauthier-Villars, JFM 63.0996.03
• Ragab, F. J. (1963). "Expansions of Kampe de Feriet's double hypergeometric function of higher order". J. f. reine angew. Mathem. (212): 113–119. doi:10.1515/crll.1963.212.113.

External links

• Weisstein, Eric W., "Kampé de Fériet function", MathWorld.